Here, we propose and demonstrate a nonlinear Purcell filter that suppresses such an undesired dephasing process without sacrificing the readout performance. When a readout pulse is applied, the filter automatically reduces the effective linewidth of the readout resonator, increasing the sensitivity of the qubit to the input field. The noise tolerance of the device we fabricated is shown to be enhanced by a factor of three relative to a device with a linear filter. The measurement rate is enhanced by another factor of three by utilizing the bifurcation of the nonlinear filter. A readout fidelity of 99.4% and a QND fidelity of 99.2% are achieved using a 40-ns readout pulse. The nonlinear Purcell filter will be an effective tool for realizing a fast, high-fidelity readout without compromising the coherence time of the qubit.